T. Tokieda Lyon, August An invitation to simple modeling of complex phenomena

Transcription

1 T. Tokieda Lyon, August 2012 An invitation to simple modeling of complex phenomena

2 Which musical note does a projec<le make on splashing into water? Of the three approaches to modeling, 3) solving the full equa<on is impossible because the full equa<on is unknown. So we must resort to 1) dimensional analysis or 2) back- of- the- envelope es<mate. We will do 2), then check it by 1).

3 2) Back- of- the- envelope es1mate What produces the sound? The projec<le creates a cavity in water. Air in the bulb acts like a spring while air in the neck acts like a mass. Let ver<cal displacement of neck air pressure difference outside/inside cavity Compressibility of air is By thermodynamics it is also where

4 Therefore the spring equa<on is mass accel force angular frequency where is the speed of sound in air.

5 Our formula passes the test of 1) dimensional analysis Take a stone of diameter. predict or E 5 (middle C is C 4 ). Blowing across the top of a boole has the same physics. predict or C 3.

6 We have been studying oscillators with forcing (<de, pendulum) and without (surface wave on water, sound), but we have not yet considered dissipa1on. We now consider phenomena involving dissipa<on (light).

7 Problem of refrac<ve index When passing through a material medium (air, water, glass,...), light appears to travel slower, at speed instead of. This factor is the refrac1ve index. It underpins Snell s law, law of reflec<on all of geometric op<cs. n 0 θ 0 θ 1 n 1 Can we es<mate from atomic data?

8 Aside : Snell s law is called loi de Descartes by the French. Pointless to debate the priority between these 17th- century gentlemen, since the law was already published by [Roger Bacon 1267] who in turn copied it from [Ibn Sahl 984]

9 How light propagates : charges oscillate electromagne<c wave (light) light makes charges oscillate Light phenomena introduce a new dimension, that of an electron charge these charges radiate new light In prac<cal problems, beoer to work with is a force, a poten<al

10 in vacuum, so consider. variables dimensions Proper<es of material medium = number of electrons per volume = mass of an electron Also = some mysterious angular frequency (of what?) Dimensional analysis yields... but let us think more carefully about this.

11 The electrons in the medium oscillate according to damping coefficient forcing by light resonant frequency of the material Hence the mysterious should in fact be. A full calcula<on reveals

12 For material made of mul<ple species of atoms, Remarkable proper<es of : is complex! Re is the refrac<ve index proper, while Im is the absorp1ve index. The graphs of Re and Im look like these : Re Im 1 resonances For radia<ons and op<cs, see

13 Wherever wave phenomena occur, there are applica<ons of refrac<on/absorp<on (op<cs, crystallography, acous<cs, seismology, tsunami defocusing...). We will see some of these, and assume for a while that is real. Note also : reflec1on can be treated as refrac1on with < 0.

14 Intermission!

15 For most materials, is in the ultraviolet of visible light. increasing func<on of blue red A prism refracts blue more than red.

16 Rainbow Water droplets in the atmosphere act as prisms. As sunlight enters a droplet it gets refracted, reflected internally, and refracted again as it exits the droplet. [Descartes 1637] Depending on whether light got reflected once, twice,... before exi<ng the droplets, we have primary, secondary,... rainbows. Observe the order of the colors.

17 Soap film In a thin layer of soap, light gets refracted, reflected internally, and refracted again as it exits the film but this <me the exi<ng light interferes also with the light reflec1ng externally.

18 Discovery of DNA double helix Refrac<on and interference occur with any frequencies of electromagne<c waves, e.g. with X- ray (but we speak of diffrac1on ). [Franklin and Gosling 1951] A `rainbow from a DNA led to the determina<on of its structure.

19 Mirage When air cools, its density increases. cool warm Below a cooler air and above a warmer ground, we have effec<vely a con<nuous stra<fica<on of prisms.

20 Sky seen from under water The whole sky is shrunk to a `porthole subtending a cone of half- angle Outside this cone, you see reflec<ons of the underwater landscape.

21 With we find

22 Čerenkov radia<on When a charge travels through a material faster than the speed of light... in that material (possible if!), a kind of shock wave is observed. [1958 Nobel prize to Čerenkov, Frank, Tamm] A blue glow in a pool of a nuclear reactor is an example, due to fast charges emioed by fission.

23 For our final example, remember that the index and that Im represents the absorp<ve index. is complex Im

24 Graph of the absorp<ve index of water It has a narrow window in which Im plunges by factor of, i.e. water passes electromagne<c waves at these frequencies but blocks them off at all others. This window is exactly the `visible spectrum. evolu1onary significance

25 Review of what we saw in lecture 3/3 sound of a splash Snell s law es<ma<ng the refrac<ve index refrac<ve index is complex, reflec<on is nega<ve refrac<on rainbow, soap film, DNA, mirage, sky from under water, Čerenkov H 2 O and evolu<on of vision

26 A Message!

27 In all natural phenomena there is something of the marvelous. There is a story which tells how some visitors once wished to meet Heraclitus, and when they entered and saw him in the kitchen, warming himself at the stove, they hesitated. But Heraclitus said, Come in; don t be afraid. There are gods even here. [Aristotle, De par1bus animalium ]

28 Don t be afraid, try simple modeling of your own phenomena. There is mathematics everywhere.

29 Don t be afraid, try simple modeling of your own phenomena. There is mathematics everywhere. Thanks for your attention